Relaxation approximation of the Euler equations

Christophe Chalons (1) , Jean-François Coulombel (2, 3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 SIMPAF - SImulations and Modeling for PArticles and Fluids

The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases.

Domaines: Equations aux dérivées partielles [math.AP]

Christophe Chalons, Jean-François Coulombel. Relaxation approximation of the Euler equations. Journal of Mathematical Analysis and Applications, 2008, 348 (2), pp.872 - 893. ⟨10.1016/j.jmaa.2008.07.034⟩. ⟨hal-01838843⟩